论文标题
后犯有限的单一政治多项式的有限属性
A finiteness property of postcritically finite unicritical polynomials
论文作者
论文摘要
令$ k $为一个数字字段，具有代数关闭$ \ bar {k} $，让$ s $是包含所有Archimedean的$ k $的有限位置。修复$ d \ geq 2 $和$α\ in \ bar {k} $，以使地图$ z \ mapsto z^d+α$在后批判性上不是有限的。假设在$α$上有技术假设，我们证明只有许多参数$ c \ in \ bar {k} $中的$ z \ mapsto z^d+c $是后有限的，$ c $是$ s $ s $ intection us s $ s $ intectial cuntection us v $ $（α）$。也就是说，在D度的单一政治多项式的模量空间中，只有许多PCF $ \ bar {k} $ - 合理点是$（（α），s），$ - 积分。我们猜想没有技术假设，同一陈述是真实的。
Let $k$ be a number field with algebraic closure $\bar{k}$, and let $S$ be a finite set of places of $k$ containing all the archimedean ones. Fix $d\geq 2$ and $α\in \bar{k}$ such that the map $z\mapsto z^d+α$ is not postcritically finite. Assuming a technical hypothesis on $α$, we prove that there are only finitely many parameters $c\in\bar{k}$ for which $z\mapsto z^d+c$ is postcritically finite and for which $c$ is $S$-integral relative to $(α)$. That is, in the moduli space of unicritical polynomials of degree d, there are only finitely many PCF $\bar{k}$-rational points that are $((α),S)$-integral. We conjecture that the same statement is true without the technical hypothesis.
